Exploring Wave-Particle Duality of Water

[Ahmad Kishki]

We know that water waves are the simple harmonic motion of particles transferring their energy along the wave, but why do we say that water waves are not just particles, given their particle origin?

Suppose then we say that a wave is just a collection of particles, then how do we explain the superposition of water waves in terms of the simple harmonic motion of each particle? Is it possible but impractical?

 

[sophiecentaur]

I'm not sure where you are going with this one; there may be a bit of confusion with the term 'particle'.. Everything 'material' is made of particles (with mass, charge etc.). Mechanical waves involve particles (molecules) bound together or, in the case of sound waves in a gas, particles bumping together (producing pressure - see Gas Laws etc). The number of particles involved is so large that gases, liquids and solids can be treated as continua. Waves are not the material itself - they are patterns of displacement of those particles. The basic calculations do not involve particles
QM uses the term 'particle' to describe the way that EM waves interact with matter in discrete packets of energy (photons). There is another sort of particle - the Phonon - which is a discrete amount of energy associated with physical vibrations. Phonons also have particle-like properties.
How many more pages did you want of this stuff? You'll have to do a lot of reading around.

 

[Ahmad Kishki]

I'm not sure where you are going with this one; there may be a bit of confusion with the term 'particle'.. Everything 'material' is made of particles (with mass, charge etc.). Mechanical waves involve particles (molecules) bound together or, in the case of sound waves in a gas, particles bumping together (producing pressure - see Gas Laws etc). The number of particles involved is so large that gases, liquids and solids can be treated as continua. Waves are not the material itself - they are patterns of displacement of those particles. The basic calculations do not involve particles
QM uses the term 'particle' to describe the way that EM waves interact with matter in discrete packets of energy (photons). There is another sort of particle - the Phonon - which is a discrete amount of energy associated with physical vibrations. Phonons also have particle-like properties.
How many more pages did you want of this stuff? You'll have to do a lot of reading around.

Where is the confusion? I am wondering why are particle and wave different phenomena if all materials are afterall made of particles? Shouldnt a more fundamental explanation of material waves then be in terms of particles?

 

[DennisN]

We know that water waves are the simple harmonic motion of particles transferring their energy along the wave, but why do we say that water waves are not just particles, given their particle origin?

It seems we are talking about mechanical waves here. The wave description captures the collective behavior of particles* (* often molecules for e.g. sound waves in air, like sophiecentaur described above, or water waves in water), and it is used because it is a useful description.

Suppose then we say that a wave is just a collection of particles, then how do we explain the superposition of water waves in terms of the simple harmonic motion of each particle?

As a result of the collective behavior of "particles" :).

Where is the confusion? I am wondering why are particle and wave different phenomena if all materials are afterall made of particles? Shouldnt a more fundamental explanation of material waves then be in terms of particles?

Again, the (mechanical) wave description is useful as a description of the collective behavior of "particles".

Please also note there are different types of waves, e.g. transverse waves and longitudinal waves.
See e.g.

• http://hyperphysics.phy-astr.gsu.edu/hbase/sound/wavplt.html#c2
• http://hyperphysics.phy-astr.gsu.edu/hbase/sound/tralon.html
  
• http://hyperphysics.phy-astr.gsu.edu/hbase/permot.html#permot

 

[sophiecentaur]

Where is the confusion? I am wondering why are particle and wave different phenomena if all materials are afterall made of particles? Shouldnt a more fundamental explanation of material waves then be in terms of particles?

So - we are talking classical waves and particles?
The treatment of the way waves behave in matter is describable in terms of bulk behaviour. But that is based on the way that individual particles interact. The Gas Laws are derived from the way individual molecules behave when they collide and the mechanical behaviour of steel wires can be related to the way molecules are 'connected' to each other. There is nothing particularly "fundamental" or worthy about discussing every wave phenomenon in terms of individual particle behaviour because it would just be pointlessly complicated. All of Science (and our lives in general, aamof) works on a series of levels because that approach gets results. You wouldn't work out how to build a bridge on the basis of individual molecular attraction. You would base your design on information about mass and strength of the materials (bulk behaviour). Civil Engineers are no less clever if they don't care to get involved with the Chemistry of the Concrete they use and the same can be said about Electronics Engineers and the quantum behaviour inside transistors.

 

[Ahmad Kishki]

So - we are talking classical waves and particles?
The treatment of the way waves behave in matter is describable in terms of bulk behaviour. But that is based on the way that individual particles interact. The Gas Laws are derived from the way individual molecules behave when they collide and the mechanical behaviour of steel wires can be related to the way molecules are 'connected' to each other. There is nothing particularly "fundamental" or worthy about discussing every wave phenomenon in terms of individual particle behaviour because it would just be pointlessly complicated. All of Science (and our lives in general, aamof) works on a series of levels because that approach gets results. You wouldn't work out how to build a bridge on the basis of individual molecular attraction. You would base your design on information about mass and strength of the materials (bulk behaviour). Civil Engineers are no less clever if they don't care to get involved with the Chemistry of the Concrete they use and the same can be said about Electronics Engineers and the quantum behaviour inside transistors.

I disagree. Science is about truth, and truth is all that concerns me here. I can not dismiss an explanation just because it is tough. If material waves are composed of particles then the particle nature is more fundamental - and i suppose a more intuitive explanation for superpositiom can be provided in terms of particles.

Yes. It is all classical waves.

 

[Ahmad Kishki]

It seems we are talking about mechanical waves here. The wave description captures the collective behavior of particles* (* often molecules for e.g. sound waves in air, like sophiecentaur described above, or water waves in water), and it is used because it is a useful description.

As a result of the collective behavior of "particles" :).

Again, the (mechanical) wave description is useful as a description of the collective behavior of "particles".

Please also note there are different types of waves, e.g. transverse waves and longitudinal waves.
See e.g.

• http://hyperphysics.phy-astr.gsu.edu/hbase/sound/wavplt.html#c2
• http://hyperphysics.phy-astr.gsu.edu/hbase/sound/tralon.html
  
• http://hyperphysics.phy-astr.gsu.edu/hbase/permot.html#permot

I never came across a model for superposition in terms of particles. So please link me to the explanation. Yes i know there are many types of waves, but this is not relevant here.

 

[sophiecentaur]

Science is about truth, and truth is all that concerns me here.

That is more Faith than Science. All Science can do is to produce models of our World and to make those models as near as possible to what is observed. That's the only sort of truth you can hope for.
Science works in 'shells' of knowledge; each layer is treated in the way that seems to work best. What you are suggesting is like demanding to know how your TV works in order to appreciate the content of a programme you are watching. Analysing what happens to light when it passes through a lens or a radio signal when it is transmitted from an antenna or the way a flute produces its note is not possible (or highly inconvenient at least) if you choose to talk in terms of particles.
Consider the fact that you assume Mathematics would be used in any analysis that you might imagine could do what you demand. Maths, itself is just a model and using it takes you outside of that 'truth' you seem to be wanting.
In my opinion, there is no serious future in your suggested approach. I feel that, if you knew more of the details of the way Science works with waves, you would realize what I am getting at. Otoh, I would be only too pleased for you to come back with a valid system for studying Science in the way you envisage.
BTW, do you have any proper references for your idea? PF policy is to require that sort of depth of knowledge of mainstream Science and to discourage unsubstantiated personal views.

 

[sophiecentaur]

I never came across a model for superposition in terms of particles. So please link me to the explanation. Yes i know there are many types of waves, but this is not relevant here.

But isn't the point of your OP that there should be one? (After all, it is a major part of wave theory)

 

[DennisN]

I never came across a model for superposition in terms of particles. So please link me to the explanation.

http://www.google.se/#q=superposition+in+terms+of+particles+-quantum

Yes i know there are many types of waves, but this is not relevant here.

I linked to those pages on HyperPhysics because I thought they would be good for you to read. Did you read them?

How about thinking about it? Let's say you take a string and fasten it to a wall at one end and grab the other end with your hand. Then you move your hand quickly up and down, producing a wave traveling along the string. What is happening to the actual string and what it's made of? What determines the amplitude of the wave? And what do we actually mean when we say "amplitude"?

 

[Ahmad Kishki]

That is more Faith than Science. All Science can do is to produce models of our World and to make those models as near as possible to what is observed. That's the only sort of truth you can hope for.
Science works in 'shells' of knowledge; each layer is treated in the way that seems to work best. What you are suggesting is like demanding to know how your TV works in order to appreciate the content of a programme you are watching. Analysing what happens to light when it passes through a lens or a radio signal when it is transmitted from an antenna or the way a flute produces its note is not possible (or highly inconvenient at least) if you choose to talk in terms of particles.
Consider the fact that you assume Mathematics would be used in any analysis that you might imagine could do what you demand. Maths, itself is just a model and using it takes you outside of that 'truth' you seem to be wanting.
In my opinion, there is no serious future in your suggested approach. I feel that, if you knew more of the details of the way Science works with waves, you would realize what I am getting at. Otoh, I would be only too pleased for you to come back with a valid system for studying Science in the way you envisage.
BTW, do you have any proper references for your idea? PF policy is to require that sort of depth of knowledge of mainstream Science and to discourage unsubstantiated personal views.

Please enlighten me, how could a question be a personal view? Thank you.

 

[Ahmad Kishki]

http://www.google.se/#q=superposition+in+terms+of+particles+-quantumI linked to those pages on HyperPhysics because I thought they would be good for you to read. Did you read them?

How about thinking about it? Let's say you take a string and fasten it to a wall at one end and grab the other end with your hand. Then you move your hand quickly up and down, producing a wave traveling along the string. What is happening to the actual string and what it's made of? What determines the amplitude of the wave? And what do we actually mean when we say "amplitude"?

I skimmed through them, i studied it all in high school. The link isn't working :/ i would add that i am asking this question to gain a deeper insight into qm, but i distanced myself from qm in the post to prevent being carried away.

Anyway, the amplitude is the particle's maximum level for shm.

What beats me is how when two waves interfere at one point, their amplitude changes bur is unaltered at any other point! I tried thinking about it (in terms of particles), but i was no where near successful. Your link didnt work, so maybe this was covered in the link. Until i find a working link. Thnx.

 

[sophiecentaur]

Please enlighten me, how could a question be a personal view? Thank you.

That is a fair point - except that you repeated the same question when an answer was given that you didn't want and you implied that 'truth' was involved.
I could ask you in what way do you think that a more 'truthful' model of classical waves could be arrived at that uses particles?

i am asking this question to gain a deeper insight into qm,

I think this could be where the confusion is arising. Quantum particles and classical particles are totally different things so the old 'particle wave duality' idea that was widely discussed in QM has no parallel in the classical world.

when two waves interfere at one point, their amplitude changes bur is unaltered at any other point!

That is not correct. The resultant of a pair of waves is a pattern that varies over the whole of space - the vectors / phasors add to produce a whole rang of resultant vectors, depending on the position is space. (The total energy has to remain the same.)

 

[Ahmad Kishki]

That is a fair point - except that you repeated the same question when an answer was given that you didn't want and you implied that 'truth' was involved.
I could ask you in what way do you think that a more 'truthful' model of classical waves could be arrived at that uses particles?

I think this could be where the confusion is arising. Quantum particles and classical particles are totally different things so the old 'particle wave duality' idea that was widely discussed in QM has no parallel in the classical world.

That is not correct. The resultant of a pair of waves is a pattern that varies over the whole of space - the vectors / phasors add to produce a whole rang of resultant vectors, depending on the position is space. (The total energy has to remain the same.)

I should have been more clear with respect to the last point: when i say the wave is unaltered in any other position except for the point of interference i was referring to the case where two non pararllel waves interfere at a point and we get interference at that point. I know the difference between quantum waves and classical waves, and this is why i was hesitant to bring up qm. Even if we consider two parallel waves interfereing then the treatment in terms of particles is still not apparent to me.

Why is a particle model more truthful? Same way qm is more truthful than cm (qm explains everything wrt the micro building blocks of matter - which is a more elementary treatment - just like what i claimed a particle model should be.) and please this is not the issue here.

The reason i repeated the question was because i felt your answer didnt address my question perfectly. I felt it was a fault of my wording, so i reworded it. I am fairly well acquainted with waves, but i never saw a treatment of wave behaviour in terms of particles, and absurdly (to me) a dichotomy between waves and particles exists very much in the classical realm, which is what i wanted to remedy.

Thank you.

 

[ZapperZ]

This is rather confusing. Where does it say that the particle model is more "truthful"? In classical physics, a wave is a wave, and a particle is a particle, and these are two different pictures. That is why it is such a "paradox" when QM came along. If it was THAT easy to reconcile those two, we won't have people who kept coming in here asking about the wave-particle duality.

But in classical physics, which is what you are restricting yourself to from your posts, these are different descriptions. Classical particles do not "interfere" or have "phases". A wave can simultaneously be in many places at once, whereas a particle can't. A wave can have a superposition of various values, while a particle cannot.

So your claim that classical particle is more "truthful" than a classical wave is puzzling. Can you look at the standard wave equation and then reduce it to a particle description? That will be the ultimate support for your claim.

Zz.

 

[Ahmad Kishki]

This is rather confusing. Where does it say that the particle model is more "truthful"? In classical physics, a wave is a wave, and a particle is a particle, and these are two different pictures. That is why it is such a "paradox" when QM came along. If it was THAT easy to reconcile those two, we won't have people who kept coming in here asking about the wave-particle duality.

But in classical physics, which is what you are restricting yourself to from your posts, these are different descriptions. Classical particles do not "interfere" or have "phases". A wave can simultaneously be in many places at once, whereas a particle can't. A wave can have a superposition of various values, while a particle cannot.

So your claim that classical particle is more "truthful" than a classical wave is puzzling. Can you look at the standard wave equation and then reduce it to a particle description? That will be the ultimate support for your claim.

Zz.

I am not making any claims, i am merely asking a question based on the assumption that everything material is made of particles. The classical wave equation itself is derived using Newton's laws i.e. based on the particle nature of the medium (we take an element of the medium and apply Newton's laws).

 

[ZapperZ]

I am not making any claims, i am merely asking a question based on the assumption that everything material is made of particles.

Actually, you did! You claim that the particle is more "truthful". This is what I questioned, and I asked you where you got that from.

If you abandoned that idea, then a classical wave description is no more "truthful" than a classical particle description, and this whole discussion is rather moot, don't you think?

The classical wave equation itself is derived using Newton's laws i.e. based on the particle nature of the medium (we take an element of the medium and apply Newton's laws).

Sorry? What "medium" is in the classical E&M description that produced the Maxwell equations? Can you provide me a reference to support your claim (again, another one) that the classical wave equation (the GENERAL form, not specific to a particular case) is derived from Newton's laws?

Zz.

 

[sophiecentaur]

It is total nonsense to suggest that two waves will cancel in just one spot and not change the amplitude elsewhere.

 

[Khashishi]

What is the smallest wave possible? Is the amplitude of the wave an integer multiple of this smallest wave? If yes, it makes sense to think of water waves as particles (perhaps). But I don't think this is the case.

 

[Ahmad Kishki]

Actually, you did! You claim that the particle is more "truthful". This is what I questioned, and I asked you where you got that from.

If you abandoned that idea, then a classical wave description is no more "truthful" than a classical particle description, and this whole discussion is rather moot, don't you think?
Sorry? What "medium" is in the classical E&M description that produced the Maxwell equations? Can you provide me a reference to support your claim (again, another one) that the classical wave equation (the GENERAL form, not specific to a particular case) is derived from Newton's laws?

Zz.

Congrats you found the exception! The wave equation in E&M is implied by maxwell's equations, but then again i wanted to talk about material waves. Please answer the question and leave aside the distractions, and exceptions.

 

[Ahmad Kishki]

It is total nonsense to suggest that two waves will cancel in just one spot and not change the amplitude elsewhere.

I think this is implied by the linearity of the wave equation. Take two waves with a single shared point, then you get a different amplitude at that point.

 

[ZapperZ]

Congrats you found the exception! The wave equation in E&M is implied by maxwell's equations, but then again i wanted to talk about material waves. Please answer the question and leave aside the distractions, and exceptions.

But that makes no sense! You have a physical concept, but somehow it makes exceptions?

Please note that you have refused to produce any reference to your claim that a particle is more "truthful" than a wave, even though I've asked for it twice. This may seem nothing to you, but it appears to be the fundamental basis to this whole thread that you created. In other words, if a particle is no more "truthful" than a wave, then you really have nothing to talk about, because your starting point is faulty.

This is neither an aside nor a distraction. It is a fundamental aspect of a logical, rational discussion. It is akin to you asking "when did you stop beating your wife?", without first establishing that the person actually (i) have a wife and (ii) was beating her up.

You have also not produce a reference to support your claim that a wave equation is derived from Newton's laws. How can we carry on a discussion on something that, to me, is faulty?

Zz.

 

[sophiecentaur]

In terms of Phonons, there is no lower limit as you could imagine a wave energy of one phonon per hour or as little as you want.
I really think you should get better acquainted with the basics of waves before trying to explore the occasional island of advanced theory in this thread

I think this is implied by the linearity of the wave equation. Take two waves with a single shared point, then you get a different amplitude at that point.

of course you get a different value but, unlike what you claim, values have to different elsewhere.

 

[Ahmad Kishki]

But that makes no sense! You have a physical concept, but somehow it makes exceptions?

Please note that you have refused to produce any reference to your claim that a particle is more "truthful" than a wave, even though I've asked for it twice. This may seem nothing to you, but it appears to be the fundamental basis to this whole thread that you created. In other words, if a particle is no more "truthful" than a wave, then you really have nothing to talk about, because your starting point is faulty.

This is neither an aside nor a distraction. It is a fundamental aspect of a logical, rational discussion. It is akin to you asking "when did you stop beating your wife?", without first establishing that the person actually (i) have a wife and (ii) was beating her up.

You have also not produce a reference to support your claim that a wave equation is derived from Newton's laws. How can we carry on a discussion on something that, to me, is faulty?

Zz.

I don't have a wife hahaha, i loved this example :D

More truthful is wrong wording (on my part) - let us say more elementary (for material waves). We know that all materials are made of particles, therefore the most elementary description of phenomena within a material are described in terms of its particles. In terms of classical physics, E&M waves are waves, yet they do not require a material, therefore they are irrelevant to my question - in fact this has been why the aether concept was introduced since for E&M "whats waving?" had no answer in 19th century physics.

The derivation for the classical wave equation is present in any waves text you can refer, as well as online. Here is the first google search result: http://www.math.ubc.ca/~feldman/m256/wave.pdf and it makes use of Newton's laws.

Now, do you see where i am getting at? If a material is made of particles and a wave exists kn the material then we (must) be able to explain this wave in terms of the particles of the medium (i think this is a logical claim) so why is there no such an explanation (at least none that i am aware of) that explains all wave phenomena in that material with respect to the particles that make up that material?

Edit: i rechecked the earlier posts, and i never really said anything about this being more truthful than that, i brought up truth only when someone implied that it may be impractical like how engineers don't mind about quantum effects. I said "more fundamental" and i invite you to go back and check.

 

[Ahmad Kishki]

In terms of Phonons, there is no lower limit as you could imagine a wave energy of one phonon per hour or as little as you want.
I really think you should get better acquainted with the basics of waves before trying to explore the occasional island of advanced theory in this thread

of course you get a different value but, unlike what you claim, values have to different elsewhere.

I was thinking of two waves perpendicular to each other, and assumed that only at the intersection would they interfere. Like z1=sin(x-vt) at y=0 as the first wave and z2=sin(y-vt) at x=0 as the second wave - they will intersect at the origin and they will only add at the origin. (I am not sure if this is a legit solution to the wave equation with each wave being constrained to two dimensions, but i suppose some boundary comdition could being about this result)

 

[ZapperZ]

More truthful is wrong wording (on my part) - let us say more elementary (for material waves). We know that all materials are made of particles, therefore the most elementary description of phenomena within a material are described in terms of its particles. In terms of classical physics, E&M waves are waves, yet they do not require a material, therefore they are irrelevant to my question - in fact this has been why the aether concept was introduced since for E&M "whats waving?" had no answer in 19th century physics.

I disagree with this. There's nothing in classical physics that says that the particle description is "the most elementary". Can you show me where this is ascribed to in any physics text?

Again, if what you are saying is true, then we would NOT be so puzzled by wave-particle duality, because, according to you, everything can be reduced to the particle picture, even in classical mechanics. Yet, the fact is that these are different phenomena in classical physics. A "wave" is a behavior governed by the wave equation. A particle isn't necessarily governed by such a behavior. Classical physics cannot describe diffraction and interference using particles!

The derivation for the classical wave equation is present in any waves text you can refer, as well as online. Here is the first google search result: http://www.math.ubc.ca/~feldman/m256/wave.pdf and it makes use of Newton's laws.

Yes? Is there more to support your claim here? I used to teach my students the same thing. I can also derive the wave equation of an oscillating mass at the end of a spring.

You are confusing different things and thinking that one is derived from the other. You also seem to have ignored an earlier comment in this thread by someone about waves being a collective phenomenon. An ocean wave cannot occur with one, two, 10, 20 water molecules. It just won't happen. Yet, you think the behavior of water molecules supersedes the behavior of the collective wave properties.

As far as I can tell, there's a lot of strange physics here.

Zz.

 

[Ahmad Kishki]

I disagree with this. There's nothing in classical physics that says that the particle description is "the most elementary". Can you show me where this is ascribed to in any physics text?

Again, if what you are saying is true, then we would NOT be so puzzled by wave-particle duality, because, according to you, everything can be reduced to the particle picture, even in classical mechanics. Yet, the fact is that these are different phenomena in classical physics. A "wave" is a behavior governed by the wave equation. A particle isn't necessarily governed by such a behavior. Classical physics cannot describe diffraction and interference using particles!
Yes? Is there more to support your claim here? I used to teach my students the same thing. I can also derive the wave equation of an oscillating mass at the end of a spring.

You are confusing different things and thinking that one is derived from the other. You also seem to have ignored an earlier comment in this thread by someone about waves being a collective phenomenon. An ocean wave cannot occur with one, two, 10, 20 water molecules. It just won't happen. Yet, you think the behavior of water molecules supersedes the behavior of the collective wave properties.

As far as I can tell, there's a lot of strange physics here.

Zz.

Okay, it seems that there is a confusion on my part, i think i was at fault. The answer is: there cannot be a particle model since a wave is a multitude of particles. Am i right?

 

[ZapperZ]

The answer is: there cannot be a particle model since a wave is a multitude of particles. Am i right?

I don't even know what "a particle model" means from your perspective! Considering that you have made up your own "truthful"ness here in this thread, I am not prepared to agree or answer such a question.

Classical particle and classical wave are two different types of descriptions in classical physics. Do you at least agree on that?

Zz.

 

[Ahmad Kishki]

I don't even know what "a particle model" means from your perspective! Considering that you have made up your own "truthful"ness here in this thread, I am not prepared to agree or answer such a question.

Classical particle and classical wave are two different types of descriptions in classical physics. Do you at least agree on that?

Zz.

Please quote my "making up of truthfulness". I gave what i saw was a logical assertion, and i was at fault and i admitted it. Before i had doubts (for materials) if they are different, i now agree. Thank you for your time.

 

[hkyriazi]

Ahmad, would you be satisfied with a justification for the use of wave equations that took the approach, "water wave A consists of X molecules with average velocity v_A1 and mass density p_A1 at position dx_A1dy_A1dz_A1 (and other velocities for all the other infinitesimal volumes of the wave) while wave B contains Y molecules with average velocity v_B1 at position dx_B1dy_B1dz_B1 etc. and whose maximum peak will intersect wave A's maximum peak at time t etc., and since no two molecules can occupy the same space they must displace each other (superpose), but since we do not have the capacity to know each and every pair of interacting molecules' exact positions and collision angles, orientations, strength of van der Waals' forces and ionic interactions (for the dipolar water molecules) etc. that impart some elasticity and variable density to the water, we shall treat the liquid medium as an incompressible, continuous medium etc., and this has been found empirically to provide an accurate description of the macroscopic behavior of intersecting waves"?

I don't mean to imply that this is exactly what a particle description would look like, but just sort of what you had in mind.

 

[Ahmad Kishki]

Ahmad, would you be satisfied with a justification for the use of wave equations that took the approach, "water wave A consists of X molecules with average velocity v_A1 and mass density p_A1 at position dx_A1dy_A1dz_A1 (and other velocities for all the other infinitesimal volumes of the wave) while wave B contains Y molecules with average velocity v_B1 at position dx_B1dy_B1dz_B1 etc. and whose maximum peak will intersect wave A's maximum peak at time t etc., and since no two molecules can occupy the same space they must displace each other (superpose), but since we do not have the capacity to know each and every pair of interacting molecules' exact positions and collision angles, orientations, strength of van der Waals' forces and ionic interactions (for the dipolar water molecules) etc. that impart some elasticity and variable density to the water, we shall treat the liquid medium as an incompressible, continuous medium etc., and this has been found empirically to provide an accurate description of the macroscopic behavior of intersecting waves"?

I don't mean to imply that this is exactly what a particle description would look like, but just sort of what you had in mind.

Yes, precisely. Thank you.